A White Paper on Using the Lasche and McKee Equations to Determine Lubricant Viscosity By MoleKule

A White Paper on Using the Lasche and McKee Equations to Determine Lubricant Viscosity

By MoleKule

Keywords: Journal bearings; lubrication.

1.0   General

Bearings reduce the friction that robs machines of horsepower. Bearings are classified by type as Roller bearings, Ball bearings, Pin/Needle bearings, Journal (sleeve) bearings, and sliding bearings. Roller and ball bearings are usually found in applications such as transmissions and differentials. Pin/Needle bearings are also found in automatic transmissions and manual transmissions that use Automatic Transmission Fluids, and in 2-cycle engines. Sliding bearings in engines can be found in cam/tappet assemblies and of course in the cylinder, where the ring pack interfaces with the cylinder liner. In this paper we will focus on the design and filtration requirements for the sleeve bearings that support journals, such as crankshafts, connecting rods, and camshafts.

1.1  Sleeve Bearing Design

A number of factors affect bearing design including loads (forces), diameter of journal/nearing, length of journal/bearing, rotational speed of journal bearing, eccentricity of journal-bearing system, clearance between journal and bearing, the minimum oil film, the viscosity of the oil film, the heat produced by all frictions, the coefficient of friction of the journal-bearing system, and the volume of oil flow.

The journal does not rotate inside the bearing in a concentric fashion, rather is has a slight wobble or “eccentricity” (denoted as “e”) that generates not only a wedge of lubricant film, but also “pumps” the oil around the wedge. Eccentricity is defined as the distance between the bearing center and the journal (shaft) center. Assume the shaft is rotating counterclockwise in the sleeve and the bearing sleeve has a hole in it, and the shaft and bearing are immersed in an oil bath. The oil will be drawn into the “diametral” clearance when the shaft is rotating and will force the wedge of oil around the inside of the sleeve. In most case, we provide both oil-in and oil-out passageways, with a higher pressure on the “in” side to keep a clean and cool oil supply flowing.

Bearing friction is determined by an empirical equation (derived by test data) called the “McKee” equation as: One uses the “McKee” equation, along with the “Lasche” equation, for bearing heat generation and dissipation, H_d and H_g, and equates these equations to find the viscosity required!

; The McKee equation

. H_g is the heat generated in a bearing, in ft-lbs/min, f is journal the coefficient of friction, D is journal diameter in in., N is journal speed in rpm, W is total radial bearing load, lbs.

; the McKee equation for journal friction. C is diametral clearance, Z is absolute viscosity of lubricating oil at temperature in centipoise, p is pressure based on projected area W/LD, in psi. N is as above. k is a constant which is derived from a graph for the L/D ratio, and is approx. 0.002  to 0.0075for most engine bearings.

; This is the Lasche equation for heat dissipated in a bearing, where  to 53 for most bearings, L is length of journal,, in in., D is as above. T_B is bearing temperature and T_O is oil temperature in ⁰F. Assume bearing temperature is 68 ⁰F (20 ⁰C) above oil temperature.

Here is the Algorithm:

1. Calculate f.
2. Calculate H_g
3. Calculate H_d
4. Set H_g = H_d and solve for Z, the viscosity.
5. Convert Z to kinematic viscosity in cSt (if desired).

We will do this for a Chevy. 350 V8 main bearing in which the bearing length L is 1.21875”, D is 2.448”, and diametral clearance averages 0.001925”. The D/C ratio is 1271.69; the L/D ratio is 0.4979. The k value for this L/D ratio is 0.006.

RPM = 3,000 rpm = N. Pressure on main bearing, p, is W/LD = 402.7 psi (W range is 700 to 1700 psi; W for the example is 1,200 psi). The bearing temperature shall be limited to 230 F and the oil temp is limited to 190 F.

Note: The algorithm presented here does overstate the viscosity by 12% but does get one into the ballpark.

 This required viscosity is between a 30 weight and a 40 weight at an oil temp of 190 ⁰F.

I.E., One takes the oil temp of 190 ⁰F and draws a vertical line to 12.9 on a Z vs. Temperature chart, the intersect shows an SAE 39 weight oil is required at the oil temps and rpm. This is about 13.0 cSt in terms of kinematic viscosity.

Shigley and Mischke, in "Mechanical Engineering Design," show that the oil flow required to maintain a 28 ⁰F differential between oil-in temp and oil-out temp of a bearing of 0.00175" clearance, is Q = 0.252 m³/s. This is for a 1.5" diameter bearing using SAE 20 weight oil at an rpm of 1800 rpm. The Chevy bearing in question is 2.448” in diameter and the length is 1.21875”.

Typical minimum film thicknesses, in hydrodynamic lubrication for these types of bearings, are on the order of 0.1 to 1um, with most operating films (in hydrodynamic lubrication) on the order of 1um to 18 um or less. Bearing clearances range from 0.0001 to 0.002 inches for every 2.54 cm (1”) of journal diameter. During normal operation the journal is supported on an extremely thin film of oil, and the two parts should have no contact. As the rotational speed increases, the lubricant is drawn into the bearing by the rotating action of the journal (shaft). Consequently, the oil film becomes thicker and the increase in friction is therefore less than directly proportional to the speed. Conversely, at lower speeds, the oil film is thinner. At extremely low speeds (such as starting or “lugging”) there is likelihood that the bearing may fail or be permanently damaged unless there is a “barrier” lubricant between the shaft and bearing that can take the load and not shear under loads.

1.2 Filtration.and Bearing Wear

In Duchowski’s paper [2], he shows that the range of most abrasive particles that cause bearing wear are in the range of 3 um to 32 um. He suggests that the filter have a beta of B >= 200 at 6 um, “Given the film thickness which exists under standard (steady-state) operating conditions, the filter element appropriate for the application should exhibit a high removal efficiency for particles down to 10 um in size. In addition, the selected filter element should also exhibit a significant removal efficiency for particle removal down to about 3 um in size to allow for thinner film which exists under startup conditions.”

References:

1. Hall, et. al., Theory and Problems of Machine Design.
2. John K. Duchowski, Examination of Journal Bearing Filtration Requirements, Lubrication Engineering, September, 1998, pp. 18-28.
3. Shigley and Mischke, Mechanical Engineering Design.